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復形和Cohen-Macaulay性質(英文版) 版權信息
- ISBN：9787030703026
- 條形碼：9787030703026 ; 978-7-03-070302-6
- 裝幀：一般膠版紙
- 冊數：暫無
- 重量：暫無
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復形和Cohen-Macaulay性質(英文版) 內容簡介
本書包含組合交換代數近幾年來的一些主要研究成果，選題圍繞單純復形、代數復形以及Cohen-Macaulay性質展開，其中的CM性質是交換代數中很為核心的研究課題。全書共分為7章。
復形和Cohen-Macaulay性質(英文版) 目錄
Contents
Preface
Notations
Chapter 1 Preliminaries on Cohen-Macaulay Rings and Modules 1
1.1 Jacobson radical and NAK Lemma 2
1.2 Modules with finite lengths 3
1.3 On graded rings and minimal graded free solution 6
1.4 Cohen-Macaulay rings and Cohen-Macaulay Modules 11
1.4.1 Dimension， height and Krull’s PI theorem 11
1.4.2 Depth Lemma and depthI(M) 14
1.4.3 Local rings: Krull dimension and a system of parameters 17
1.4.4 Cohen-Macaulay modules and Cohen-Macaulay rings: local case 22
1.4.5 Cohen-Macaulay rings: non-local case 26
1.4.6 Cohen-Macaulay rings: graded case 28
1.4. 7 Gorenstein rings 34
1.5 Sequentially Cohen-Macaulay modu1es 35
Chapter 2 Abstract Simplicial Complexes 40
2.1 Definitions， fundamiental properties and examples 40
2.1.1 Abstract simplex and abstract simplicial complex＊40
2.1.2 0ther notations and symbols on a simplicial complex ＊ 41
2.1.3 Fundaｍental operations on sub-complexes and geometric realization of an abstract simplicial complex 42
2.2 The facet idea1I(*) and Stanley-Reisner ideal I*of a simplicial complex * 47
2.2.1 Monomial ideals and ideal operations 47
2.2.2 The Stanley-Reisner (nonface） ideal I* and facet ideal I* 47
2.2.3 The Alexander dual simplicial complex * of*and related properties 48
2.2.4 Square-free monomial ideal I: its nonface complex， facet complex; I* and f-ideals 54
2.3 Relative simplicial complexes and relative nonface ideals 55
Chapter 3 Shellable Simplicial Complexes 57
3.1 Destriction and examples 57
3.2 Restriction maps and Rearrangement Lemmas 60
3.3 (r，s)-skeleton * 64
3.4 Shifted， vertex-decomposable and shellable conditions for a simplicial complex 66
3.5 Shellable and k-decomposable 69
Chapter 4 Chain Complex Reduced from a Simplicial Complex and Koszul Complexes 73
4.1 The chain complex reduced from an abstract simplicial complex and reduced homology groups 74
4.2 Koszul complexes of lengths 1 or 2 79
4.3 Koszul complexes of geueral length 80
4.3.1 Exterior algebra constructed from a module 80
4.3.2 Koszul complexes： two commonly used definitions 81
4.4 Koszul complexes: a S田nmary of main results 83
4.5 Other resolutions and complexes of monomial ideals 84
4.5.1 The Taylor resolution 84
4.5.2 The Scarf complex 86
4.5.3 The Lyubeznik resolutions 89
Chapter 5 (Sequentially) Cohen-Macaulay Simplicial Complexes and Graphs 91
5.1 Cohen-Macaulay simplicial complexes 92
5.1.1 Fundamental properties and characterizations 92
5.1.2 Connected in codimension one 97
5.1.3 Minimal Cohen-Macaulay simplicial complexes and shelled over 100
5.2 Matroid complexes 104
5.3 Pure shellable， constructible， and Cohen-Macaulay 107
5.4 A graded ideal with linear quotients and shellable complexes 111
5.4.1 A graded id兇i with linear quotients 111
5.4.2 Shellable complexes and monomial ideals having linear quotients 115
5.4.3 Powers of edge ideals of graphs and regularity 118
5.4.4 A polymatroidal monomial ideal has linear quotients 119
5.4.5 Strongly shellable simplicial complexes 120
5.5 sCM simplicial complexes and sCM graded modules 121
5.6 Clique complex *， edge ideal I(G) and cover ideal Ic(G) 123
5. 7 Vertex-decomposable graphs a且d shellable graphs 124
5.8 Minimal verex covers and standard irredundant primary decomposition of I(G) 127
5.9 Cohen-Macaulay graphs and well-covered graphs 129
5.10 Shellable clutters 131
5.10.1 Clutters with the free vertex property 132
5.10.2 Chordal clutters 133
5.11 Some particular classes of graphs 133
5.11.1 Bipartite graphs 133
5.11.2 Boolean graphs are Cohen-Macaulay 138
5.11.3 Cactus graphs and classes of vertex-decomposable graphs 143
5.11.4 Cameron-Walker graphs 146
5.11.5 Chordal graphs 147
5.11.6 F-simplicial complexes and f-ideals of kind （n，d） 150
5.11.7 Gap-free graphs and related H-free graphs 174
5.11.8 Graphs whose complements are r-partite 176
5.11.9 Graph expansions and graph blow ups 185
5.11.10 Interlacing graphs * and triangular graphs * 188
5.11.11 Vertex clique-whiskered graphs * and their generalizations * 189
5.11.12 1-decomposable graphs 202
Chapter 6 Shellable Simplicial Complexes from Posets 204
6.1 Preliminaries 204
6.2 A bounded， locally upper-semimodular poset is pure shellable 205
6.3 EL-labeling of a poset and EL-shellable graded posets 209
6.4 Admissible lattices and SL-shellable poset 213
6.5 CL-shellable poset and recursive atom orderings 215
6.5.1 Rooted interval and CL-shellable poset 216
6.5.2 Recursive atom orderings 218
Chapter7 Betti Numbers and Castelnuovo-Mumford Regularity 222
7.1 Calculating Betti numbers via the functor Tor 222
7.2 Polarization keeps the Betti numbers and regularity unchanged 224
7.3 Hochster’s Formula and other two reformulations 226
7.4 Graded Betti numbers of graphs: some general reults 230
7.5 Spli也table
Preface
Notations
Chapter 1 Preliminaries on Cohen-Macaulay Rings and Modules 1
1.1 Jacobson radical and NAK Lemma 2
1.2 Modules with finite lengths 3
1.3 On graded rings and minimal graded free solution 6
1.4 Cohen-Macaulay rings and Cohen-Macaulay Modules 11
1.4.1 Dimension， height and Krull’s PI theorem 11
1.4.2 Depth Lemma and depthI(M) 14
1.4.3 Local rings: Krull dimension and a system of parameters 17
1.4.4 Cohen-Macaulay modules and Cohen-Macaulay rings: local case 22
1.4.5 Cohen-Macaulay rings: non-local case 26
1.4.6 Cohen-Macaulay rings: graded case 28
1.4. 7 Gorenstein rings 34
1.5 Sequentially Cohen-Macaulay modu1es 35
Chapter 2 Abstract Simplicial Complexes 40
2.1 Definitions， fundamiental properties and examples 40
2.1.1 Abstract simplex and abstract simplicial complex＊40
2.1.2 0ther notations and symbols on a simplicial complex ＊ 41
2.1.3 Fundaｍental operations on sub-complexes and geometric realization of an abstract simplicial complex 42
2.2 The facet idea1I(*) and Stanley-Reisner ideal I*of a simplicial complex * 47
2.2.1 Monomial ideals and ideal operations 47
2.2.2 The Stanley-Reisner (nonface） ideal I* and facet ideal I* 47
2.2.3 The Alexander dual simplicial complex * of*and related properties 48
2.2.4 Square-free monomial ideal I: its nonface complex， facet complex; I* and f-ideals 54
2.3 Relative simplicial complexes and relative nonface ideals 55
Chapter 3 Shellable Simplicial Complexes 57
3.1 Destriction and examples 57
3.2 Restriction maps and Rearrangement Lemmas 60
3.3 (r，s)-skeleton * 64
3.4 Shifted， vertex-decomposable and shellable conditions for a simplicial complex 66
3.5 Shellable and k-decomposable 69
Chapter 4 Chain Complex Reduced from a Simplicial Complex and Koszul Complexes 73
4.1 The chain complex reduced from an abstract simplicial complex and reduced homology groups 74
4.2 Koszul complexes of lengths 1 or 2 79
4.3 Koszul complexes of geueral length 80
4.3.1 Exterior algebra constructed from a module 80
4.3.2 Koszul complexes： two commonly used definitions 81
4.4 Koszul complexes: a S田nmary of main results 83
4.5 Other resolutions and complexes of monomial ideals 84
4.5.1 The Taylor resolution 84
4.5.2 The Scarf complex 86
4.5.3 The Lyubeznik resolutions 89
Chapter 5 (Sequentially) Cohen-Macaulay Simplicial Complexes and Graphs 91
5.1 Cohen-Macaulay simplicial complexes 92
5.1.1 Fundamental properties and characterizations 92
5.1.2 Connected in codimension one 97
5.1.3 Minimal Cohen-Macaulay simplicial complexes and shelled over 100
5.2 Matroid complexes 104
5.3 Pure shellable， constructible， and Cohen-Macaulay 107
5.4 A graded ideal with linear quotients and shellable complexes 111
5.4.1 A graded id兇i with linear quotients 111
5.4.2 Shellable complexes and monomial ideals having linear quotients 115
5.4.3 Powers of edge ideals of graphs and regularity 118
5.4.4 A polymatroidal monomial ideal has linear quotients 119
5.4.5 Strongly shellable simplicial complexes 120
5.5 sCM simplicial complexes and sCM graded modules 121
5.6 Clique complex *， edge ideal I(G) and cover ideal Ic(G) 123
5. 7 Vertex-decomposable graphs a且d shellable graphs 124
5.8 Minimal verex covers and standard irredundant primary decomposition of I(G) 127
5.9 Cohen-Macaulay graphs and well-covered graphs 129
5.10 Shellable clutters 131
5.10.1 Clutters with the free vertex property 132
5.10.2 Chordal clutters 133
5.11 Some particular classes of graphs 133
5.11.1 Bipartite graphs 133
5.11.2 Boolean graphs are Cohen-Macaulay 138
5.11.3 Cactus graphs and classes of vertex-decomposable graphs 143
5.11.4 Cameron-Walker graphs 146
5.11.5 Chordal graphs 147
5.11.6 F-simplicial complexes and f-ideals of kind （n，d） 150
5.11.7 Gap-free graphs and related H-free graphs 174
5.11.8 Graphs whose complements are r-partite 176
5.11.9 Graph expansions and graph blow ups 185
5.11.10 Interlacing graphs * and triangular graphs * 188
5.11.11 Vertex clique-whiskered graphs * and their generalizations * 189
5.11.12 1-decomposable graphs 202
Chapter 6 Shellable Simplicial Complexes from Posets 204
6.1 Preliminaries 204
6.2 A bounded， locally upper-semimodular poset is pure shellable 205
6.3 EL-labeling of a poset and EL-shellable graded posets 209
6.4 Admissible lattices and SL-shellable poset 213
6.5 CL-shellable poset and recursive atom orderings 215
6.5.1 Rooted interval and CL-shellable poset 216
6.5.2 Recursive atom orderings 218
Chapter7 Betti Numbers and Castelnuovo-Mumford Regularity 222
7.1 Calculating Betti numbers via the functor Tor 222
7.2 Polarization keeps the Betti numbers and regularity unchanged 224
7.3 Hochster’s Formula and other two reformulations 226
7.4 Graded Betti numbers of graphs: some general reults 230
7.5 Spli也table
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復形和Cohen-Macaulay性質(英文版) 節選
Chapter 1 Preliminaries on Cohen-Macaulay Rings and Modules All rings in this book are assumed to be commutative with identity element and， all modules are unital. An f.g. module RM denotes that M is finitely generated as a left R-module. In this section， we collect some fundamental results of commutative algebra and， efforts are made to make the exposition self-contained. We take the monograph [39] (in English）as the fundamental reference in commutative algebra， and one can also use [119] (in Chinese）for some basic facts of commutative rings. This section also contains two particular topics. Recall that a ring R is called Cohen-Macaulay (often abbreviated as CM for brevity) if the local ring Rp is CohenMacaulay，i.e， depth( Pp) = dim (Rp) holds for every prime (or equivalently， maximal) ideal P of R. For any field ，set and . Then for any graded proper ideal I of S， there is a single condition on R =: S/ I， i.e.， depth(m/1，R) = dimR， which we call grnded Cohen-Macaulay. In Therorem 1.61， we will give a direct proof to the following expected facts: The graded ring S/ I is graded Cohen-Macaulay if and only if it is Cohen-Macaulay. In the final part， we include an elementary approach of Schenzel’s view on sequentially Cohen-Macaulay modules， in which the key tool is the saturated submodule of nM related to an ideal I of R，see [39， Section 3.6] for a general discussion on the importance of the local cohomology module . 1.1 Jacobson radical and NAK Lemma For a ring R， the Jacobson radical J(R) is defined to be the intersection of all maximal left (or equivalently， right) ideals of R. Note that J ( R) contains no nonzero idempotent elements of R， and where U(R) consists of all units of R. Lemma 1.1 (NAK Lemma) Let M be an f.g. module over α ring R. For an ideal I of R，if and IM= M，then M = 0. Proof Let and let . It follows from that α ＝ Aα for some . Then we have where and， denotes the adjoint matrix of a matrix B. Since 1 + b is invertible in R，It follows that ， thus M = 0. Corollary 1.2 Let M be an f. g. module over a ring R and let N be a submodule of M. For an ideal I of R， if and ， then. Thus the module J(R)M is a superfluous (or alternatively， a small) submodule of an f.g. module M. Note that Nakayama’s Lemma follows easily from the following Cayley-Hamilton Theorem: Theorem 1.3 Let I be an ideal of R， and assume that α module RM is generated by n elements. Then for each endomorphism of M with， there exist elements in such that the polynomial annihilates ． Proof For any ，set xm ＝ .Assume RM= . Then there exists a matrix such that ， where .Since R is assumed to be commutative， we have where denotes the adjoint matrix of Bin Mn(R). Then the polynomial annihilates*， where * holds for all i. Cayley-Hamilton Theorem can be applied to verify the following: Corollary 1.4 Let R be a commutative ring and RM an f.g. module. Then ( 1) Any surjective endomorphism of M is an isomorphism. (2) A commutative ring has the Invariance Basis Nwmber， i. e.， any module isomorphism implies n = m. Note that a non.commutative Noetherian ring also has the properties in Corollary 1.4. The proof to this fact is left as an exercise. Remark NAK is the abbreviation of Nakayama-Azumaya-Krull. It is also the first three letters of Nakayama. 1.2 Modules with finite lengths Recall that a module RM has finite length if and only if M is both Artinian and Noetherian. For a prime ideal P of R， recall that if and only if ann . Recall that Supp(P) consists of prime ideals P such that . The following proposition implies the well-known Jordan-Holder Theorem: Lemma 1.5 Let (l.2.1) be a composition series of the module M， i.e.， each quotient module is simple. Let Then (1) For any maximal ideal Q of R ， if and only if . Thus Supp(M) and hence， it is independent of the choice of the composition series. (2) For distinct maximal ideals P and Q of R， we have *. (3) For any *， the number of Mi such that * is the length of Mp as an Rp-module. Proof (1) Let * and assume*Then we have *and *. Take* such that *. Then we have*， hence*. Thus *， and it follows from * that MQ = 0. On the other hand， if *， assume* for some i. Since R/Q is a field， we have*. Thus*， hence*. This shows*， and completes the proof to (1). Note that * holds for any maximal ideal P distinct to Q. (2) and (3): If M is a simple module，then*， where P = ann(M). Then Mp = M and MQ = 0 for any maximal ideal *. Now assume length(M) = n > 1. Then for any maximal ideal P*A， we have an Rp-module sequence: Note that each M3/M3-1 is a simple module， and while the latter holds if and only if*. Thus the length of the Rpmodule Mp is the number of Mk such that*. In particular， it follows from the filtration (1.2.2) that (Mp )Q = 0 holds for distinct maximal ideals P and Q. Theorem 1.6 Let M be a module with
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